Syllabus for Geometry Qualifying Exam

The Geometry 1 exam covers essentially the material taught in MTH G122 (Geometry 1).

Topics

• Differentiable manifolds, vector bundles, vector fields and differential equations, Frobenius theorem.
• Differential forms, Stokes’ theorem, deRham cohomology.
• Riemannian metrics, geodesics, completeness.
• Connections on vector bundles, curvature.
• Hypersurfaces in R n Gauss-Bonnet theorem, Jacobi fields.
• Lie group and Lie algebras.

References

• Michael Spivak, A Comprehensive Introduction to Differential Geometry, 3rd Edition, Vol. I and II, Publish or Perish, 1999
• Noel J. Hicks, Notes on Differential Geometry, Van Nostrand, 1965.